Nonlinear elliptic equations with critical potential and critical parameter

2006 ◽  
Vol 136 (5) ◽  
pp. 1041-1051 ◽  
Author(s):  
Yao Tian Shen ◽  
Yang Xin Yao
Keyword(s):  
Elliptic Equation ◽  
Hardy Space ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Critical Parameter ◽  
Nonlinear Elliptic Equations ◽  
Point Theory ◽  
Positive Answer ◽  
Critical Potential ◽  
Nonlinear Elliptic

We give a positive answer to an open problem about Hardy's inequality raised by Brézis and Vázquez, and another result obtained improves that of Vázquez and Zuazua. Furthermore, by this improved inequality and the critical-point theory, in a k-order Sobolev–Hardy space, we obtain the existence of multi-solution to a nonlinear elliptic equation with critical potential and critical parameter.

Nonlinear Elliptic Equations in Strip-Like Domains

2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Jaeyoung Byeon ◽  
Kazunaga Tanaka
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic

AbstractWe study the existence of a positive solution of a nonlinear elliptic equationwhere k ≥ 2 and D is a bounded domain domain in R

scholarly journals MULTIPLE CONDENSATIONS FOR A NONLINEAR ELLIPTIC EQUATION WITH SUB-CRITICAL GROWTH AND CRITICAL BEHAVIOUR

2001 ◽  
Vol 44 (3) ◽  
pp. 631-660 ◽  
Author(s):  
Juncheng Wei
Keyword(s):  
Elliptic Equation ◽  
Critical Points ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic Equations ◽  
Subject Classification ◽  
Critical Behaviour ◽  
Unknown Constant ◽  
Secondary 35J40 ◽  
Nonlinear Elliptic

AbstractWe consider the following nonlinear elliptic equations\begin{gather*} \begin{cases} \Delta u+u_{+}^{N/(N-2)}=0\amp\quad\text{in }\sOm, \\ u=\mu\amp\quad\text{on }\partial\sOm\quad(\mu\text{ is an unknown constant}), \\ \dsty\int_{\partial\sOm}\biggl(-\dsty\frac{\partial u}{\partial n}\biggr)=M, \end{cases} \end{gather*}where $u_{+}=\max(u,0)$, $M$ is a prescribed constant, and $\sOm$ is a bounded and smooth domain in $R^N$, $N\geq3$. It is known that for $M=M_{*}^{(N)}$, $\sOm=B_R(0)$, the above problem has a continuum of solutions. The case when $M>M_{*}^{(N)}$ is referred to as supercritical in the literature. We show that for $M$ near $KM_{*}^{(N)}$, $K>1$, there exist solutions with multiple condensations in $\sOm$. These concentration points are non-degenerate critical points of a function related to the Green's function.AMS 2000 Mathematics subject classification: Primary 35B40; 35B45. Secondary 35J40

scholarly journals FOUNTAIN-LIKE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS WITH CRITICAL GROWTH AND HARDY POTENTIAL

2005 ◽  
Vol 07 (06) ◽  
pp. 867-904 ◽  
Author(s):  
VERONICA FELLI ◽  
SUSANNA TERRACINI
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Blow Up ◽  
Nonlinear Elliptic Equations ◽  
Critical Growth ◽  
Hardy Potential ◽  
Nonlinear Elliptic ◽  
Hardy Type ◽  
Type Potential

We prove the existence of fountain-like solutions, obtained by superposition of bubbles of different blow-up orders, for a nonlinear elliptic equation with critical growth and Hardy-type potential.

On Existence of L∞-Ground States for Singular Elliptic Equations in the Presence of a Strongly Nonlinear Term

2007 ◽  
Vol 7 (3) ◽  
Author(s):  
J.V. Goncalves ◽  
A.L. Melo ◽  
C.A. Santos
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Term ◽  
Ground States ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic ◽  
Singular Elliptic Equations

AbstractWe establish new results concerning existence and the behavior at infinity of solutions for the singular nonlinear elliptic equation −Δu = ρa(x)u

scholarly journals Regularity of Extremal Solutions to Nonlinear Elliptic Equations with Quadratic Convection and General Reaction

2020 ◽  
Vol 17 (6) ◽  
Author(s):  
Asadollah Aghajani ◽  
Fatemeh Mottaghi ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Dirichlet Boundary ◽  
Nonlinear Elliptic Equations ◽  
Extremal Solution ◽  
General Reaction ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Regularity Results ◽  
Increasing Function

AbstractWe consider the nonlinear elliptic equation with quadratic convection $$ -\Delta u + g(u) |\nabla u|^2=\lambda f(u) $$ - Δ u + g ( u ) | ∇ u | 2 = λ f ( u ) in a smooth bounded domain $$ \Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N ($$ N \ge 3$$ N ≥ 3 ) with zero Dirichlet boundary condition. Here, $$ \lambda $$ λ is a positive parameter, $$ f:[0, \infty ):(0\infty ) $$ f : [ 0 , ∞ ) : ( 0 ∞ ) is a strictly increasing function of class $$C^1$$ C 1 , and g is a continuous positive decreasing function in $$ (0, \infty ) $$ ( 0 , ∞ ) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution $$u^*$$ u ∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function $$ h(t)=f(t)e^{-\int _0^t g(s)ds}$$ h ( t ) = f ( t ) e - ∫ 0 t g ( s ) d s , nor that the functions $$ gh/h'$$ g h / h ′ or $$ h'' h/h'^2$$ h ′ ′ h / h ′ 2 admit a limit at infinity.

scholarly journals An oscillation theorem for the nonlinear degenerate elliptic equation in the Heisenberg group

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Duan Wu ◽  
Pengcheng Niu
Keyword(s):  
Elliptic Equation ◽  
Heisenberg Group ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Degenerate Elliptic Equation ◽  
Oscillation Theorem ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
Oscillation Of Solutions ◽  
Nonlinear Degenerate

AbstractThe aim of this paper is to study the oscillation of solutions of the nonlinear degenerate elliptic equation in the Heisenberg group $H^{n}$ H n . We first derive a critical inequality in $H^{n}$ H n . Based on it, we establish a Picone-type differential inequality and a Sturm-type comparison principle. Then we obtain an oscillation theorem. Our result generalizes the related conclusions for the nonlinear elliptic equations in $R^{n}$ R n .

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